Scattering of Phonons by Charge Carriers in Superfluid Helium: The Zero-Velocity Limit

The problem of phonon scattering by charge carriers in superfluid helium is considered, with the aim of interpreting recent measurements of charge-carrier mobilities below 1 °K. The calculations are done in the hydrodynamic approximation, and explicitly include the effect of the electrostrictive variations in the fluid surrounding the core structures of the carriers. For the positive carrier, the necessary core properties are calculated, and in particular it is found that the core does not act like a hard sphere. The theoretically predicted phonon-limited mobility for positives is found to be in excellent agreement with experiment, provided a liquid-solid surface energy of order 0.1 erg ${\mathrm{cm}}^{-2\mathrm{}}$ for the core is assumed. In the case of the negatives, results are similar to those of previous workers. The mobility of the electron bubble is substantially explained by resonance scattering of phonons, but a significant discrepancy remains which probably arises from the use of the idealized bubble model. The changes in the theoretical mobility curves that can be produced by varying the bubble parameters $a_{-}$ and $V_1$ are of the order of this systematic discrepancy, so that the best values of these parameters cannot be determined from the data. Values of $a_{-}$ less than 15 Å can, however, be ruled out with some confidence. The roton contribution to the scattering is determined by subtracting the phonon part, and turns out to have some unexpected features. For the positives, $\frac{e}{{\mu }_{+}}\cong 1.34\times {10}^{-9\mathrm{}}{T}^{-\frac{1}{2}}{e}^{-\frac{\Delta }{\mathrm{kT}}}$. The temperature dependence of $\frac{e}{{\mu }_{-}}$ is not as clearly established, but the prefactor lies between $T^0$ and $T^{-\frac{1}{2}}$. The ratio of $\frac{e}{{\mu }_{-}}$ to $\frac{e}{{\mu }_{+}}$ is found to have a surprisingly small value of about 1.5, perhaps indicating that electrostrictive variations in the fluid surrounding the positive core scatter rotons strongly. No present theory of roton scattering explains the observed features.